int_0^(pi/2)sin^5thetacosthetad theta=?

Explanation:

Another Explanation (5): সমাধান: ধরি, \(I = \int_{0}^{\frac{\pi}{2}} \sin^5(\theta) \cos(\theta) \, d\theta\) এখানে, \(\sin(\theta) = u\) ধরি। তাহলে, \(\cos(\theta) \, d\theta = du\) যখন \(\theta = 0\), তখন \(u = \sin(0) = 0\) যখন \(\theta = \frac{\pi}{2}\), তখন \(u = \sin(\frac{\pi}{2}) = 1\) সুতরাং, সমাকলনটি \(u\) এর সাপেক্ষে পরিবর্তন হয়ে দাঁড়ায়: \(I = \int_{0}^{1} u^5 \, du\) \(I = \left[ \frac{u^6}{6} \right]_{0}^{1}\) \(I = \frac{1^6}{6} - \frac{0^6}{6}\) \(I = \frac{1}{6} - 0\) \(I = \frac{1}{6}\) অতএব, \(\int_{0}^{\frac{\pi}{2}} \sin^5(\theta) \cos(\theta) \, d\theta = \frac{1}{6}\) 🎉